The underworld city of Nexus has a current population of 500,000 people. The city’s population is growing exponentially with a growth rate of 3% per year. The city’s infrastructure can support a maximum population of 1 million people. At what time does the city need to upgrade its infrastructure to accommodate the growing population? Model the equation of it first and then its derivative

Solution

T h e u n d er w or l d c i t y o f N e xu s ha s a c u rre n t p o p u l a t i o n o f 500, 000 p eo pl e . T h e c i t y ’ s p o p u l a t i o n i s g ro w in g e x p o n e n t ia ll y w i t h a g ro wt h r a t e o f 3% p er ye a r . T h e c i t y ’ s in f r a s t r u c t u re c an s u pp or t a ma x im u m p o p u l a t i o n o f 1 mi ll i o n p eo pl e . A t w ha t t im e d oes t h e c i t y n ee d t o u p g r a d e i t s in f r a s t r u c t u re t o a cco mm o d a t e t h e g ro w in g p o p u l a t i o n ? M o d e l t h e e q u a t i o n o f i t f i rs t an d t h e n i t s d er i v a t i v e

Amount after addition : 515000

Solution steps

Translate the problem into an equation:

x = 500000 + 0.03 \cdot 500000

x = 500000 + 0.03 \cdot 500000 : x = 515000

Amount after addition : 515000